Design Platform for Planar Mechanisms Based on a Qualitative Kinematics

Design Platform for Planar Mechanisms based on a Qualitative Kinematics

Bernard Yannou Adrian Vasiliu Laboratoire Productique Logistique Ecole Centrale Paris 92295 Chatenay-Malabry, France phone +33 141131285 fax +33 141131272 email [yannou I adrian]@cti.ecp.fr

Abstract: Our long term aim is to address the open problem of combined structural/dimensional synthesis for planar mechanisms. For that purpose, we have developed a design platform for linkages with lower pairs which is based on a qualitative kinematics. Our qualitative kinematics is based on the notion of Instantaneous Centers of Velocity, on considerations of curvature and on a modular decomposition of the mechanism in Assur groups . A qualitative kinematic simulation directly generates qualitative trajectories in the form of a list of circular arcs )out and line segments. A qualitative simulation amounts to a n slider-crank propagation of circular arcs along the mechanism from the Figure 1 : Example of a linkage, composed of a mechanism (bodies 1, 2 and 3) and a dyad (bodies 4 and 5). Joints A motor-crank to an effector point. A general understanding and G are revolute joints (rotation) connected to the fixed frame. of the mechanism, independent of any particular Joint D is a prismatic joint connected to the fixed frame. The dimensions, is extracted . This is essential to tackle other joints are revolute joints connecting two bodies. There is a synthesis problems . Moreover, we show that a motor atjoint A, so body 1 is a crank dimensional optimization of the mechanism based on our qualitative kinematics successfully competes, for the path a requirement of desired kinematics in the form of the desired trajectory effector point, generation problem, with an optimization based on a of a point, termed on conventional simulation. a body of the mechanism, making abstraction of the orientation of this body. This problem is called the path generation problem. 1 Introduction A synthesis process consists of choosing the type of Mechanical linkages with lower pairs are planar mechanism (type synthesis), e.g., slider-crank or four-bar mechanisms of fixed topology, consisting of rigid bodies mechanism, and the optimal dimensions for this mechanism constantly connected with joints of revolute type (allowing (dimensional synthesis). The dimensional synthesis a rotation) or prismatic type (allowing a translation). An problem has been widely studied for two decades and more example is given in Fig. 1 . Linkages with lower pairs [13; 25], and some satisfactory techniques have been represent a large part of the existing mechanisms in implemented. However, the combined type/dimension industry. Our subject is to develop an efficient automatic or synthesis problem remains an open problem for two major semi-automatic method for the synthesis of these linkages. reasons. On one hand, it often consists in testing a huge Such a method should find one or several optimal number of simple equivalent mechanisms in terms of a mechanisms submitted to functional requirements. These unique criterion: the number of degrees of freedom [12; 24; requirements are conventionally: 26] . On the other hand, the criteria which preside over the pruning of such a combinatory are completely " the desired kinematics of one or several bodies, questionable. In fact, the choice of the type of mechanism is " no collision between the mechanism and its very little constrained by the requirements and, for a given environment, type of mechanism, there is no classification of the trajectories in terms of the dimensions. This reveals that at " the minim;zation of the efforts in joints when the two previous criteria are fulfilled. present we have very poor and unstructured knowledge about the interactions between functions, structures and For the moment, we have limited our research to: dimensions for the mechanisms. " linkages of one-degree-of-freedom, It is for this reason that we developed a platform for the design of linkages which is based on a qualitative

kinematics. The idea of this qualitative kinematics is to In the case of the generation of qualitative explanations, consciously accept a small loss in numerical accuracy of the exact trajectory is generated by a primaryconventional trajectories, in order to handle more global and qualitative kinematic simulation for specific dimensions of the information and to have a deeper understanding of mechanism which must be explained . This is the trajectory mechanisms . This is made by directly generating qualitative of the effector point which intervenes in the specification, trajectories in the form of lists of circular arcs and line i.e., in the function of the mechanism, so that we can say segments, instead of generating lists of points like in a that the sought function drives the explanations . The conventional simulation . process of the generation of the qualitative explanations is Firstly, we show that a dimensional optimization summed up in Fig. 2- (synthesis) process based on (looping over) such a qualitative kinematic simulation is faster than an optimization process based on a conventional simulation. For brevity, we will term these two dimensional optimization processes : qualitative optimization and exact (or conventional) optimization . Secondly, and this is the real challenge, we show how to construct a qualitative understanding of a mechanism as an engineer could do, in order to justify the qualitative shape of the trajectory from particular dimensions . Only the work of Shrobe [27] deals with this purpose but in a less explicit manner . The qualitative explanations that our system generates use two Figure 2: Generating qualitative explanations of the behaviour of a mechanism . mechanisms of causal deductions: a modular decomposition of the mechanism and a causal graph of Instantaneous In the case of qualitative optimization (which is a loop Center of Velocityl calculations. Thirdly, we hope that, in over a qualitative kinematic simulation), the exact the long term, we shall be able to invert the explanation trajectory is specified by the designer himself because this process . Then, we would be able, in a combined is the desired trajectory to fulfill . After a qualitative type/dimension synthesis process, to intelligently index dimensional optimization, optimal dimensions for the some mechanism types which are serious candidates and to mechanism are proposed : this is the qualitative optimum directly propose value intervals for the dimensions . (see Fig. 3) . In order to confirm the validity ofour approach Section 2 presents the two functionalities of the design (in spite of the two consecutive approximations : first platform : generation of the qualitative explanations and approximation into arcs and segments and qualitative qualitative optimization . We evoke, in section 3, the simulation), we will have to compare the qualitative modular decomposition of a mechanism and its advantages . optimum with the exact optimum directly obtained by a In section 4, we present our qualitative kinematics. Section conventional optimization . 5 shows how our qualitative kinematics can be applied to a dimensional optimization . The most important section is 'Mechanism type' section 6 where we show how to construct qualitative WITHOUT any explanations for a mechanism. Finally, in section 7 we specific dimension compare our work with others on qualitative kinematics. Approximation 2 The two functionalities of the design EXACT into arcs and QUALITATIVE desired traj. platform segments d6sired traj .

2.1 Qualitative explanations and qualitative dimensional optimization

Both functionalities of the design platform (generation of VALIDATION QUALITATIVE optimum the qualitative explanations and qualitative optimization) require as a first step an approximation of an exact Figure 3: Qualitative dimensional optimization. The method trajectory (in the form of a list of points) into a qualitative remains to be validated by comparison between the qualitative trajectory (in the form of a list of arcs and segments) . In and the exact optima . both cases, this resulting qualitative trajectory is used to drive a qualitative kinematic simulation. 2.2 Approximation of an exact trajectory into arcs and segments

This approximation takes as input argument a maximum 1 Henceforward, they are termed ICs. admissible error. This error is the maximum distance

Figure 4: Approximation of an exact trajectory (left) defined by 48 points into a qualitative trajectory (right) defined by 4 circular arcs. between a point of the exact trajectory and the qualitative trajectory. A minimization of the total number of arcs and segments is carried out with a least square algorithm . For the same total number of arcs and segments, the maximum number of segments is preferred because they have more Figure 6: A systemic representation of the linkage of Fig. 1 by a significant qualitative properties . Finally, the third modular decomposition into multipole groups. criterion is to prefer the approximation which minimizes considered as a sub-system, receives from the previous the error. The algorithm does not respect the continuity group information about the position, velocity and between the arcs (and segments) because it is not necessary. acceleration of its input joints. After an internal An example of approximation is given in Fig. 4. calculation according to the transfer function of the group, the kinematic features of the output poles are determined 3 Modular decomposition of linkages and transmitted to the next group. Modular decomposition of linkages is an efficient Thus, a configuration resolution can be viewed as a approach for linkage analysis and a way to better motion propagation from the fixed frame to the terminal understand a mechanism [4;10;17]. It consists in dividing group. In normal situations, the terminal group contains the the mechanism into elementary structural groups, called effector point in order for each group to contribute to the Assur groups. An Assur group is obtained from a kinematic position and velocity of the effector point. Otherwise, the chain of zero mobility by suppressing one or more links, at superfluous groups must certainly have another function in the condition that there is no simpler group inside. the mechanism and the motion of their internal poles will Theoretically, the number of Assur groups is not limited, be interesting for future qualitative explanations. This is but practically only the simplest are used. the case of the linkage in Fig. 1 with an effector point in E; An interesting extension of the theory of Assur groups point F (see Fig. 6) is detected as an interesting point for consists in a systemic view of the modular approach. explanations. Pelecudi [20; 21] extended the notion of Assur group to the The transfer function is known a priori for each type of concept of a multipole group as an elementary sub-system Assur group. For most of the simpler groups, the position, possessing input poles (joints), motion and effort transfer velocity and acceleration of the output poles are expressed functions, internal poles and output poles (see Fig. 5). Input in an explicit manner from the kinematic features of the and output poles are considered as potential joints, input poles. Such a modular decomposition is a systematic allowing connection with other groups. We have approach to reduce iterative calculations for any type of implemented in our platform an algorithm for the automatic linkages. It can be considered as a symbolic compilation of modular decomposition of linkages. This decomposition is the mechanism, depending only on its type and not on its unique (see Fig. 6). particular dimensions. It allows a kinematic simulation A kinematic simulation consists in looping over a method which is more efficient than computing and solving configuration determination corresponding to a particular equations for the whole mechanism at once. It can help crank angle. During a configuration resolution, each group, kinematic (and also kinetostatic) analysis, but also synthesis (for more details, see [30]). In our platform, the modular decomposition will be used e for a resolution in positions (not in velocities) inside a , ALN t ~ C ` a ¬ o~ qualitative kinematic simulation.

o 4 The qualitative kinematic save simulation The qualitative kinematic simulation is the basis of our Figure 5: Examples of the simplest Assur groups and the work. In very few iterations (corresponding to the number corresponding systemic icons.

of arcs and segments), it generates a "rough" simulation in terms of numerical accuracy but its interest is double :

" to obtain a good elementary calculus time inside a dimensional optimization process,

to build qualitative explanations of mechanism behaviour .

This qualitative simulation is based on the notions of Instantaneous Centers of Velocity (ICs) and some notions of curvature in plane kinematics. Figure 7: Calculation of the effector point velocity vector. Graphically, the method is simple: the velocity vector of 4.1 An approach based on the Instantaneous M and vector a have the same norm, the velocity vector of the Centers (ICs) effector point T and vector c too . Vectors. a, b and c are orthogonal to the straight line of the ICs. During the movement of a mechanism, it can be considered that the relative movement between any couple We have developed an algorithm which provides the of bodies (say i and j) is locally2 equivalent to a rotation method to determine the unknown ICs from the known ICs of center IC;I . This is the Instantaneous Center of Velocity by using the theorem ofthree centers and with a minimum of between the bodies i and j. Of course, if bodies i and j are intermediary ICs . According to the theorem of three centers, connected by a revolute joint, IC;I is constantly the center two known ICs : 1Cil ,and ICIk define the straight line of the joint, fixed relatively to body i and body j. where lies ICik , termed AlCik. In like manner, a known IC: Otherwise, IC;I changes over time . In like manner, a IC& and a direction Dij coming from a prismatic joint, prismatic joint between bodies i and j, is kinematically define a straight line AICik . The elementary step of the equivalent to a revolute joint whose the relative IC is at algorithm is the determination of a new IC lCik by infinity in a direction Dy orthogonal to the direction of intersection of two known straight lines AlCik . translation . Traditionally, the ICs are used in graphical methods to calculate a specific absolute velocity (relative Let us take the mechanism of Fig. 1 and suppose that the to the fixed frame 0) or a relative velocity between two positions of the joints are known. Hence, six ICs are bodies, starting from the known velocity of the motor- already known from a total of fifteen (C6 ): ICON is joint A, crank as in the one-degree-of-freedom mechanism of Fig. 1. 'Cl 2 is joint B, IC23 is joint C, IC24 is joint E, IC45 is joint A remarkable property of these ICs is given by the theorem F, ICo5 is joint G. The effector point is E and an interesting of three centers: point is F (see also §6) . These points are both fixed to body The relative ICs of any three bodies: ICij , ICIk and 4. We saw that we were to determine ICMT and ICOT , i.e. IC ik lie on the same straight line. in our case : IC14 and ICp4 . For this case, only one intermediary IC turns out to be necessary: 1C02 . The Now, let us consider a particular position the motor- of algorithm proceeds in two steps: building first a causal crank for which the rotational velocity is known. Let us graph of IC deductions (see Fig. 8.a) and, from this graph, denote M a point on the motor-crank, hence its velocity generating a symbolic trace of IC deductions (see Fig. 8.b) . vector is supposed to be known. The IC between the fixed This trace respects the sequentiality of the IC calculations, frame (0) and the motor-crank (M, too), denoted ICOM , is i.e., at any moment the calculation of a new IC is carried constantly the center of the fixed revolute joint. Let us out from previously known or calculated ICs. The method denote T, standing for trajectory, the effector point and, by of IC determination can be graphically displayed to the extension, the body where it is attached . The knowledge, for a particular crank position, of the positions of the effector point and ICs: 1CMT and ICpT , leads to the calculus of the velocity vector of the effector point relatively to the fixed frame, as explained in Fig . 7. These two interesting ICs are not a priori known.

It is worth knowing that when ICMT and 1COM are geometrically equal, the effector point has a zero velocity ICo2 = (ICo1, ICI2)n ,IC23 (it locally stands still) . (Dos IC04 =(ICo2 ,IC 24 )n(ICp5 ,IC45 )

(b) IC14 =(ICo1 ,ICo4 )n(ICi2 ,IC24 )

Figure 8: An IC causal graph (a) and the corresponding symbolic 2 "Locally" means during an infinitesimal period oftime. trace (b) which shows the order of IC calculations .

joint D

Figure 9: Method of determination of the sought ICs for the mechanism ofFig. 1. designer (see Fig. 9). This algorithm of IC determination is applied once for a mechanism type and a given effector Figure 10: Moving and fixed centrodes. point; it is independent of any specific dimensions. The of three centers. Presently, we do not think that symbolic trace of deductions is used for automatically explanations for understanding the mechanism could be generating the coordinate equations of the sought ICs and easily extracted. Instead, we use a numerical correction the velocity vector equations of points E and F (as which is presented further . But, for the purpose of explained in Fig. 7) in a symbolic form. In this manner, explanation generation, some concepts are worth studying velocity vectors are calculated without any iteration and further: by the call to an optimized procedure. Like for the modular the inflection circle which is the location of effector decomposition, it can be said that the mechanism is on body T whose curves have a segment, 1.compilated". The fact that there is no iteration for the points determination ofvelocity in a general case of mechanism is the cubic of stationary curvature which is the location a good result from the point of view of speed. Even the of effector points whose curves really contain a modular approach of Assur groups does not escape to circular arc. iterative calculations of the velocity for some (rare) Assur groups. 4.3 Propagation of circular arcs along the mechanism 4.2 Curvature of qualitative arcs and centrodes Now, let us examine the problem of the correspondence Our goal is not to calculate the velocity vector of the between the limiting points of the arcs and segments of the effector point from the rotational velocity of the motor- desired qualitative trajectory of T, and the positions of the crank, but it is to directly obtain a circular arc (or a motor-crank M. Let us take the example of Fig. 11 where segment) centered on a position of the effector point and the approximation of the exact trajectory gives one segment corresponding to a finite rotation of the crank. The center and three arcs. and the radius of curvature the trajectory of the effector of The four corresponding positions of the crank, which point is thus needed at this particular position. But the have to be determined, are parameterized by the angle 00 center of curvature not the Instantaneous of this arc is corresponding to the initial position 0 and three angles Center of Velocity ICoT , as we could expect. Indeed, ICoT 00i (i varying from 1 to 3), relatively to this position. is locally the center of relative rotation, but the following instant 'COT has moved. We can have a better In the case of the generation of qualitative explanations, understanding in considering the loci of the ICs in the fixed the exact desired trajectory is provided by a primary frame (0), namely thefixed centrode, and the loci of the ICs conventional simulation. Therefore, it is possible to in the frame of body T, namely the moving centrode . By represent the curvilinear abscissa of the points of the exact rolling the moving centrode attached to body T about the desired trajectory function of the relative angle of the fixed centrode, the motion of body T is faithfully motor-crank. An interpolation between these points gives represented (see Fig. 10). It is clear that the IC: ICOT , the the function ST = f(Breiative ) (see Fig. 12). The angles Bp; center of curvature and the effector point are aligned. are straightforwardly determined from the curvilinear Problems of curvature in planar motion have been abscissas sTi of the limiting points of the arcs and widely studied by mathematicians during the eighteenth segments. and nineteenth centuries [31] . The radius of curvature of In the case of a dimensional optimization, there exist two the arc can be calculated by the Euler-Savary equation types of path generation problem according to the [13; 25]. Although the Euler-Savary equation can be solved specification: by a graphical method, this is not a purely qualitative theory like the deductions of ICs by the use of the theorem

0,4

Figure 11 : What is the correspondence between the limiting Figure 13: (a) Arcs and segments of the qualitative trajectory of points of the arcs and segments of the desired qualitative the effector point T. (b) Arcs of the qualitative trajectory of point trajectory of T and the positions of the motor-crank ? M attached to the motor-crank.

" The effector point must move along the desired From now, we will consider, at the inverse, a qualitative trajectory in accordance with prescribed temporal kinematic simulation where the circular arcs of the current positions ofthe crank. It amounts to impose the curve trajectory of the effector point must be calculated from the sT = f(B,elarive~~ i.e. angles 0oi corresponding to sT . circular arcs of the motor-crank . For each position of the If this curve is the straight line ((0,0),(2tr,ST,. )), it motor-crank in M; (the mid-point of a circular arc), a means that the curvilinear velocity of the effector resolution of positions is carried out, using the graph of point and the angular velocity of the motor-crank are modular decomposition ; the positions of the effector point proportional. The designer can also define areas of T; and the positions of the joints are then determined. The relatively high acceleration or deceleration for the velocityvector of the effector point V(Ti) is determined by effector point. the specific procedure generated from the symbolic trace of The effector point must move along the desired IC deductions . This vector gives the orientation and the trajectory whatever the corresponding positions of length of the arc centered on Ti . Indeed, benefiting from the the crank may be. Thus, angles do and Boi become fact that the ratio of the curvilinear velocities are constant design variables which are to be valued during the on an arc, the curvilinear total lengths of the two optimization process. This represents the general corresponding arcs respect the same ratio. Thus, we case. directly consider that the velocity values are equal to the curvilinear length values. At present, the determination of Now, let us consider the mid-points M; of the circular the radius of curvature is solved by a second position arcs of the trajectory of M (see Fig. 13.b) and the mid-points resolution of the mechanism very close from the mid-point Ti of the circular arcs and line segments of the qualitative of the arc Ti . It can be seen in Fig. 10 that this second desired trajectory (see Fig. 13.a). position is sufficient because we already know the straight In saying that points T temporally correspond to points line (ICOT ,Ti ) where the center of curvature lies. Mi, we commit a piecewise linear approximation of the The deduction of the velocities from the motor-crank to curve sT = f( B,eiative (compare Fig. 14 and Fig. 12). In fact, the effector point through multiple ICs can be viewed as a the effector is considered to move with a constant velocity propagation of a velocity vector through the mechanism. along a circular arc (for a constant rotational velocity of Therefore, since a velocity vector and a circular arc are the motor-crank). Again, this acceptable approximation somewhat equivalent (except for the radius of curvature), confirms that we construct a qualitative kinematics. sT sT sTaux sTmax sT3 ST3

sT2 sT2 sTi T1 0relative 0relative 0 0 0 M1 001 002 003 2n 0 001 002 003 27t Figure 14: The piecewise linear approximation which is made Figure 12: The knowledge of the curve ST = f( e,eiative leads because of the use of qualitative arcs and segments (to be to the determination of angles 9oi . compared with the original curve in Fig. 12).

the image of the propagation of circular arcs from the trajectories, only the final curves are then compared. In motor-crank to the effector point can be adopted. [11], for example, the calculated trajectory is interpolated by a spline for each simulation and is compared to the 5 Dimensional optimization based on spline interpolation of the desired trajectory. The penalty qualitative kinematics function between the two trajectories is the sum of squares of two sets of points equally spaced along each spline. But, In this section, our concern is to dimensionally optimize these interpolations are extremely costly. Therefore, in our a given mechanism for the path generation problem, i.e. to qualitative approach, we prefer to have, for the sake of find the best dimensions in order for an effector point to efficiency, a few additional design variables for the crank move along a desired trajectory. angles and to perform a one-to-one comparison between the Several types of methods are currently used: graphical two lists of arcs and segments. The cost due to the distance techniques [13; 25], direct analytical methods [2] and between the desired and the simulated trajectories takes optimization techniques (non-linear programming) [l; 3; 4; into account the coordinates of the limiting points A and C 11; 18; 24; 28; 32] . A major drawback in using analytical of the arcs and segments and their mid-points B, according methods is that the number of prescribable points3 of the to the formula: desired trajectory is limited by the type of mechanism. 12 2 ['((XAd 5 (YAd YAS) Therefore, for a large class of practical design problems, 2 -X'4 + ) +,' we need optimization methods, which represent a general w1 Bd - XBs )2 (YBd - YBs) ) + design tool. However, this is a hard problem especially Foist Nb . arcs rcs (( X + 2 because of its high non-linearity. ,2 )2 (I XCd -XCs +(YCd -YCS ) After having specified an exact desired trajectory, the 2 designer approximates it into a qualitative desired The factor 1 for the limiting points comes from the fact trajectory (let us take the example of Fig. 4). Next, the 2 designer chooses a mechanism type (the one of Fig. 1) and that these points are counted twice, in two neighbour arcs specifies the effector point (E) and the design variables or segments. Even if they are not identical because of the which can vary in the optimization process. By default, the non-continuity, they correspond logically to the same point coordinates of revolute joints and the angles of prismatic on the desired trajectory, so they must be counted only joints, corresponding to the initial configuration of the once. wt is a weight coefficient. mechanism, are design variables. The coordinates of the effector point are not design variables because the effector The optimization algorithm must penalize the point is constrained to be at the position 0 of the desired mechanisms which encounter blocking positions during a trajectory. The bar lengths are deduced from these simulation. For that purpose, we propose a penalty term variables. Moreover, angles eo and Bp i of the motor- which is proportional to the square of the difference crank, corresponding to the qualitative desired trajectory, between the curvilinear length of the desired trajectory (Ld ) and the curvilinear length of the simulated trajectory must be added, when no temporal constraint is specified on (Ls). the motion of the effector point. In the example of Fig. 1, it When, during a simulation, a blockage is encountered, can be deduced from the modular decomposition of Fig. 6 the simulation is rerun backwards in order to obtain the that the dyad group composed by bodies 4 and 5 does not maximum value for Ls . So, we have a penalty term: intervene in the motion of the effector point E. The general Fblock = w2 (Ld -Ls )2 . Finally, the term Fconstr is design vector is then: introduced in order to take into account user defined (XA , YA , XB,YB,XC,YC,aD,Bo,eot,e02,eo3) constraints such as: the linkage must never cross a delimiting box, one bar length must be lower than a value, Here, an important advantage of the qualitative and so on. So, the global penalty function is: approach is that the number of variables of angles ( eo and 9oi ) is equal to the number of qualitative arcs (and F = Fdlst +Fblock + Fconstr segments). For a conventional approach, this number We adopted an optimization algorithm based on the would have been equal to the number of points of the gradient method because of the non-accessibility to partial desired trajectory. In the example of Fig. 4.b and 4.a, these derivatives, the non-linearity and inequality constraints . numbers are respectively 4 and 48. Anyway, the choice of the optimization algorithm turns out In fact, for a conventional approach with numerous to be almost independent from the type of kinematic precision points, another technique is used to avoid this simulation (qualitative or not) . For the example of Fig. 1 design variable explosion. The aim is to make abstraction (11 design variables), the use of a qualitative simulation of any temporal correspondence between the two instead of a conventional simulation speeds up the optimization, practically with a factor of two, which corresponds the speed ratio between two simulations of 3 We find also in the literature: precision points. different type. Indeed, instead of dealing with a description

15), we notice that body 5 stands still during a rotation of the motor-crank of about 140°; this is the dwell phenomenon. 90 This is due to the fact that when the effector 87 point is moving along arc A3 (see Fig. 16), the length of body 4 is 81 such that joint F is constantly at the center of curvature of 7s arc A3. Our system does not only describes this 69 phenomenon but it also explains why it occurs for the 62 particular dimensions of the mechanism. This is an

u essential difference with the workby Shrobe.

49 Joint F was, for us, an interesting point. When we try to 86 72 198 144 188 216 a2 see 124 369 determine the velocity of joint F, for the position Figure 15: The curve of the angle of body 5 (see Fig. 1) corresponding of the mid-point of arc A3: T3 , we fall on the as a function of the angle of the motor-crank exhibits a particular case (see Section 4.1) where /C45 and IC04 are period of constant angle named a dwell. geometrically equal, i.e. the distance is lower than a fixed of the desired trajectory by a collection of points, admissible distance. Thus, we have proved symbolically qualitative simulation deals with a much smaller number of that body 5 stands still locally for the position arcs and segments while at the same time almost faithfully corresponding to T3 . This is a necessary condition for reproducing the exact curve when the mechanism reaches body 5 to stand still for a period of time (dwell) . The ideal dimensions. Finally, we have confirmed, on several second necessary condition is that the center of curvature examples, the validity of our model of qualitative is also geometrically identical IC45 (joint F), which is optimization in finding a qualitative optimum very close to immediately shown. Concerning the first condition, our theexact optimum system is capable of generating many more relevant explanations, in explaining why, in this particular 6 Understanding linkages position, the dimensions of the mechanism caused /C45 and The title of this section is deliberately the same as an IC04 to be geometrically equal. According to the causal article by Shrobe [27]. Below, we will compare both graph of ICs, we know that 1C04 is obtained by: approaches of understanding linkages. Presently, let us ICo2 =(1Co1 ,IC12)"') (Do3 ,IC23) take again the mechanism of Fig. 1. The effector point is joint E and its exact trajectory is given in Fig. 4.a. This ICo4 =(/Cp2 ,IC24) n (ICo5 , IC45) exact trajectory is approximated into four circular arcs With IC45 and IC04 identical, the system infers, after with a high degree of accuracy (see Fig. 4.b). The mechanism exhibits the same phenomenon as the mechanismof Shrobe's some symbolic handling, that the three straight lines article, namely a dwell. Indeed, observing the curve of the (1Co 1,1C12 ), (IC45 ,/C24 ) and (D03,lC23) are concurrent at angle of body 5 as a function of the crank angle (see Fig. the same point ICo2 . This result is presented graphically to the designer which understands that: " the bar of body 1, the bar of body 4 and a vertical bar passing through joint C are necessarily concurrent " (see Fig. 16). From particular dimensions of a mechanism of a given type, we have extracted a general understanding of this mechanism and we have given dimensional conditions to respect in order to perform the dwell function . This last feature is not implemented by Shrobe. However, this generalization of the knowledge linking type, dimensions and functions is an essential condition to efficiently tackle the problem of combined type/dimension synthesis. 7 Related works Our work addresses both the domain of linkage synthesis and the domain of qualitative-kineiiatics.- We already stated (in the introduction), that the recent works on a combined type/dimension synthesis of linkages [24; Figure 16: Our system exhibits graphically the necessary 28] were not satisfactory because of the poorness of the condition to have a dwell-mechanism: three straight lines knowledge about the interactions between mechanical attached to bodies must be concurrent.

functions, mechanism types and mechanism dimensions. specific dimensions of the mechanism. Anyway, we claim This is why we developed a qualitative kinematics to that there is no conjecture to be made, because all the extract general explanations on the mechanism behaviour relationships between an arc of the desired trajectory, the from particular dimensions. corresponding arcs of the motor-crank and other interesting points, and the particular dimensions of the The two primary major approaches in qualitative mechanism, are given from the ICs chaining and kinematics were those of Joskowicz & al [14;15;16;19; 22; considerations of curvature . The consequences in our and Faltings & al [5; 6; 7; 8; 9]. The two approaches are 23] understanding of a mechanism is that Shrobe explains that very similar in principle : there is a dwell (see section 6) because joint E revolves they are dedicated to mechanisms of changing around joint F on a circular arc of appropriate radius, topologies, with contacts of complex shapes, whereas we explain why this circular arc exists (because " the configuration space of a mechanism is defined by three straight lines related to the mechanism are combining both quantitative and symbolic concurrent) and because we relate dimensions to kinematic information, functions. " a qualitative kinematic simulation consists in 8 Conclusion forecasting the state transitions, i.e. changes in the topology (a contact is established or broken). We have presented a new qualitative kinematics for mechanical linkages, which is based on a deep knowledge On one hand, Joskowicz & al consider three-dimensional of the planar kinematics: considerations of Instantaneous mechanisms with fixed axes; on the other hand, Faltings & Centers of Velocity and of centers of curvature. A of al consider two-dimensional (or planar) mechanisms qualitative kinematic simulation amounts to propagate a ratchet 7], or two degrees of freedom like a pawl and [6; circular arc of trajectory from the motor-crank to an clocks [9] . effector point, thanks to an IC causal graph. It permits to In [7], Faltings and Sun began to explore the extract a general understanding of the mechanism from a relationships between the part dimensions and the specific simulation. This understanding relates for a kinematic functions. Given a mechanism and some initial mechanism type the dimensions to the kinematic functions dimensions, theypropose a technique to redesign the system (e.g., a dwell) . This knowledge relating type (structure), with new dimensions in order to meet functional dimensions and functions is essential to tackle the requirements. combined type/dimension synthesis, which is still an open Subramanian and Wang [29] defined a synthesis tool of problem for linkages. We already showed that the use of a three-dimensional, fixed topology, single-degree offreedom qualitative kinematic simulation in a dimensional problem) mechanisms. This tool proposes to assemble elementary optimization process (for the path generation mechanisms of fixed axes in order to meet global kinematic successfully competes with a conventional approach. requirements. Moreover, we know that our qualitative kinematics But these works are dedicated to distinct classes of theory is extensible to: mechanisms from that of our work: the linkages (planar " several effector points and corresponding desired mechanisms of fixed topology) with lower pairs . trajectories, Consequently, they can not really be compared with ours. " rotational motors not related to the fixed frame, Only the work of Shrobe [27] which we mentioned " translational motors, above deals with linkages. Shrobe does not develop a qualitative kinematics. He just considers the result of a several degrees of freedom, conventional kinematic simulation, i.e. the trajectories of " gear pairs. interesting points, that he approximates into circular arcs and line segments. The interesting points are, principally, Our future work will consist in exploiting the the input/output points of building blocks, sorts of modular qualitative explanations of linkages into a general groups as ours. He detects some similarities between the synthesis system. features of the qualitative curves (e.g. same interval of time for two arcs). His concern is to construct an understanding Acknowledgements of a mechanism in conjecturing causal relationships Thanks to Mounib Mekhilef and Andrei Sebe for their between these qualitative features. Because his model does assistance in optimization algorithms, and to Alan Swan not include deep knowledge, like our qualitative for comments on earlier drafts of this paper. kinematics based on the ICs and curvature considerations, his explanations will remain hypothetical and incomplete. Moreover, we are convinced that conjectures can only generate judicious explanations in very rare cases, for

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